Except the cost, I don't see a problem.
I'd still do a quick math of how many hours you want to fly and then compare shipping cost to bring and return the helicopter, insurance cost, import tax/vat if you stay longer than 180 days in the EU and all that to the hour rate if you rent a helicopter at your destination. Europe is expensive, but I've done the import myself and it's not gonna be cheap. Of course if you rent, you might have to get your license converted first, which means learning, a couple of tests and a checkride, plus paying for all of that.
There are lots of N registered ships in Europe, also owned by non-Americans (who need an American friend or a trust company managed by an American national to be on the register).
Make sure you have a service center or some other mechanic available where you are planning to fly. You will still need FAA approved folks to sign off your annual inspection if I am not wrong (100hr should be fine).
I'd say the easiest would be UK if you do things by yourself, because of the language. Otherwise you could speak to a company to handle things for you. They will have to take the rotor blades off in the US and put them back on in Europe anyway... not sure what your make of helicopter is, but if it's an R22 I know people in Switzerland and Germany that can help.
If you relocate to Europe, you might be able to bring your helicopter without paying tax. At least if you'd bring your own car it would be that way. The Denmark route is another possibility without pay tax.